The beating heart creates blood pressure and flow pulsations that propagate as waves through the arterial tree that are reflected at transitions in arterial geometry and elasticity. Waves carry information about the matter in which they propagate. Therefore, modeling of arterial wave propagation extends our knowledge about the functioning of the cardiovascular system and provides a means to diagnose disorders and predict the outcome of medical interventions. In this review we focus on the physical and mathematical modeling of pulse wave propagation, based on general fluid dynamical principles. In addition we present potential applications in cardiovascular research and clinical practice. Models of short- and long-term adaptation of the arterial system and methods that deal with uncertainties in personalized model parameters and boundary conditions are briefly discussed, as they are believed to be major topics for further study and will boost the significance of arterial pulse wave modeling even more.